Lebesgue proved that every separately continuous function f: R×R®R is a pointwise limit of continuous
functions. W. Rudin extended this by showing that if X is a metric
space, then for any topological space Y, every separately continuous
function f: X×Y®R is a pointwise limit of
continuous functions. This statement can fail if we take for X an
arbitrary linearly ordered space, even if X is separable. However, we
show that if X = X1×¼×Xn, where X1,...,Xn are
linearly ordered spaces which are either all separable or all countably
compact, and Y is any topological space, then every separately
continuous function f: X×Y®R is Borel
measurable. We also give, under a cardinal arithmetic assumption, an
example of a linearly ordered space X and a separately continuous
function f: X×X®R which is not Borel measurable.

The famous `Finite sums theorem' proved by N. Hindman in 1970's says
that for every partition of the integers into finitely many pieces
there is an infinite set A of integers such that all finite sums of
distinct elements of A belong to the same piece.

This result has been extended in various ways to partitions of other
`Ramseyan structures'. A typical result says that a given
combinatorial space S has the property that for every partition of
S into finitely many pieces S has a large homogeneous
substructure.

I will talk about theorems of Hales-Jewett and Gowers and their various
extensions, both true and false.

Recent years have seen an increasing use of model theory in
Set-theoretic Topology. I will discuss some results that were obtained
using model-theoretic methods; these include existence of universal
spaces and uniqueness results for certain spaces. I will also discuss
some questions that arose out of this work but for which it is not
clear whether they require topology or model theory (OR JUST GOOD OLD
FASHIONED insightful analysis and intricate combinatorics) to solve
them.

A classification of maximal almost disjoint (MAD) families of
subsets of a countable set along the lines of Rudin-Keisler ordering of
ultrafilters is presented. A connection with a problem of J. Roitmann
as to whether an existence of a dominating family of size À1
implies the existence of a MAD family of size À1 is also
discussed.

Let RanPowers be the expansion of the ordered field of real
numbers by the restricted analytic functions and the power functions.
Then RanPowers is a polynomially bounded (o-minimal)
reduct of (Ran,exp), the expansion by restricted analytic
functions and the exponential function (cf. [M]). L. v. d. Dries
conjectured that RanPowers is a maximal
polynomially bounded reduct of (Ran,exp).

Let Â be an elementary extension of (Ran,exp), and
let ÂanPowers be the corresponding reduct. Let
H(Â) be the Hardy field of germs at infinity of R-definable (1-variable) functions. Then H(RanPowers) is a Hardy subfield of H(Â). In this
talk, we show that H(ÂanPowers) enjoys the following
maximality property:

In particular this shows that H(ÂanPowers) is maximal
among those Hardy subfields of H(Â) arising from polynomially
bounded reducts of Â. This gives a partial answer to the above
conjecture. The proof of the maximality property is based on a
structure theorem for H(Â) established in [K-K] and [KS].

It has been shown that MAÀ1 is equivalent to a Ramsey
theoretic statement about partitions of finite subsets of w1.
Replacing ``finite'' with ``n-element'' leads to an array of formally
weaker axioms whose exact relationship to MAÀ1 is currently
unknown. This talk will examine some of the consequences of the 2 and
3 dimensional versions of Martin's Axiom.

The model completion of the theory of differential fields can be
axiomatized by the Frobenius Theorem of differential geometry. In
other words: Let T be the theory of fields of characteristic zero
with several commuting derivations. Then we can augment T by axioms
saying the following, that a collection of differential one-forms have
a common zero, provided that the differential ideal that they generate
is the same as the ordinary ideal. The result is the model completion
of T.

We will consider stable structures definable in o-minimal models. Our
conjecture is that Zilber's principal holds for these structure. As the
first step we will consider fields of finite Morley Rank and show that
these fields must be pure.

If an observer moves steadily away from the Cayley graph of a finitely
generated group, then any finite configuration will eventually become
indistinguishable from a single point; but he may observe certain
finite configurations which resemble earlier configurations. The
asymptotic cone is a topological space which encodes all of these
recurring finite configurations. Unfortunately the construction of an
asymptotic cone involves the choice of a nonprincipal ultrafilter on
the set of natural numbers, and it was not clear whether the resulting
asymptotic cone depended on the choice of the ultrafilter. In this
talk, answering a question of Gromov, I shall present an example of a
finitely generated group which has two non-homeomorphic asymptotic
cones. This is joint work with Boban Velickovic.